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This paper presents a proximal bundle variant, namely, the relaxed proximalbundle (RPB) method, for solving convex nonsmooth composite optimizationproblems. Like other proximal bundle variants, RPB solves a sequence of proxbundle subproblems whose objective functions are regularized compositecutting-plane models. Moreover, RPB uses a novel condition to decide whether toperform a serious or null iteration which does not necessarily yield a functionvalue decrease. Optimal iteration-complexity bounds for RPB are established fora large range of prox stepsizes, both in the convex and strongly convexsettings. To the best of our knowledge, this is the first time that a proximalbundle variant is shown to be optimal for a large range of prox stepsizes.Finally, iteration-complexity results for RPB to obtain iterates satisfyingpractical termination criteria, rather than near optimal solutions, are alsoderived.

A proximal bundle variant with optimal iteration-complexity for a large range of prox stepsizes